Fermat's Last Theorem - the Polynomial Version

Pierre de Fermat (1601 −1665) stated his “Last Theorem” that xⁿ + yⁿ = zⁿ has no nontrivial solutions in the integers for n ≥ 3 in the margin of his copy of Diophantus’s Arithmetica in 1637. In one of the boldest claims by one of the brightest individuals in the history of mathematics, Fermat wrote that he had a proof, but that he did not have enough room to write it in the margin. It is very likely that his proof was incomplete. Nonetheless, his innocent enough statement incited hundreds of capable (and not so capable) individuals into feverish work for over three and one-half centuries. These individuals made great accomplishments in mathematics; the development of Modern Algebra being one of the foremost. This intriguing chapter in mathematics’ history came to a close in 1995 with the work of Andrew Wiles and Richard Taylor.

We will present a proof of a version of Fermat’s Last Theorem for Polynomials. In developing the proof of this result, the important open Number Theory problem known as the abc Conjecture will be presented and we will discuss Shinichi Mochizuki's claim of proving the problem.  Also discussed will be Waring's Problem and the beautiful result know as the Mason-Strother's Theorem of which I had the honor of being asked to contribute an article to the Pi Mu Epsilon Journal's Centennial Edition's on 100 Math Milestones.

Though the proof involves some topics from abstract algebra, the audience will be reminded of basic definitions with the talk being accessible to anyone that understands derivatives.  We note that the proof of the polynomial version of Fermat’s Last Theorem is significantly shorter than the integer version.